It turns out that he was known almost exclusively as Buster in his early years. Here is a clear reference from The Meridian Daily Journal, August 19, 1931 when Buster was just approaching the record for consecutive games played:

There are literally over a thousand references to Buster Gehrig in Google’s archived newspaper reports of the era. I hope we can get this nickname added to his Baseball Reference page, it seems a shame that such a great moniker for a hungry young up-and-comer, as he was in his early years, could be lost to history.